Mathematical Physics

The research work of this group involves statistical and asymptotic properties of classical and quantum systems; the theory of integrable systems and exactly solvable quantum field and statistical mechanics models; random matrices and orthogonal polynomials; modern theory of special functions; representation theory of Lie algebras and quantum group; noncommutative geometry; quantum chaos; along with related number theoretic aspects of spectral theory.

The Mathematical Physics Group maintains very strong working contacts with all the leading world centers in the Group's research areas. The members of the Group have been repeatedly invited as short-term visiting scholars by several top Universities, and the Group has hosted several high-profile visitors-Bernard Malgrange, Tetsuji Miwa, Sergei Novikov, just to name a few. The Group also intends to strengthen its potential by inviting long-term visitors for carrying out important joint research projects

Paul Bleher

Prof. Bleher's principal research interests focus in four directions: statistical physics, quantum integrable systems, theory of random polynomials, and random matrix models. Bleher is a world expert in the rigorous theory of phase transitions in systems with random interactions and in the relations between the quantum spectra of integrable quantum models and the properties of the underlying systems of classical mechanics. His long-term collaborators in this area include Jean Ruiz and Valentin Zagrebnov (Center for Theoretical Physics, Luminy-Marseille, France), Freeman Dyson and Jean Bourgain (Institute for Advanced Study, Princeton), Joel Lebowitz (Rutgers University), and Yakov Sinai and Denis Kosygin (Princeton University).

The theories of random polynomials and random matrix models represent Bleher's most recent areas of research. Together with Bernard Shiffman and Steven Zelditch (of Johns Hopkins University), he solved the problem of universality and scaling for systems of random multivariate polynomials or, more generally, random sections of powers of line bundles over a compact Kahler manifold. This problem has significant applications to the theory of quantum chaos.

On random matrices, Pavel Bleher has been collaborating with Alexander Its, the other member of the Mathematical Physics Group, since both of them came to the Department in 1993-1994. In 1995, Bleher and Its launched a joint project on the Riemann-Hilbert approach to the study of the universalities in the theory of random matrix models and orthogonal polynomials. This approach allowed them to solve several long-standing problems concerning the asymptotic analysis of the orthogonal polynomials with exponential weights (e.g., the so-called Nevai problem). Simultaneously, these works of Bleher and Its have been followed by a flurry of activity in the area with an increasing involvement of new researchers with very different backgrounds and experiences, some of whom have already become Bleher's new collaborators.

Alexander Its

Prof. Its' major area is integrable systems. His current research interests are concentrated in the following directions: (a) Asymptotic analysis of the matrix models and orthogonal polynomials via Riemann-Hilbert and isomonodromy methods; (b) The asymptotic analysis of the correlation functions of quantum exactly solvable models and the related aspects of the theory of Fredholm and Toeplitz operators; (c) The theory of integrable nonlinear partial and ordinary differential equations of the KdV and Painleve types.

In the area of random matrices, Its' main results for the last seven years have been obtained in collaboration with Pavel Bleher, as described above.

The asymptotics of correlation functions has been a major theme of Its' research for the last 20 years. The most recent of Its' efforts in the area of correlation functions are concerned with the problem of evaluation of quantum entanglement in the XY spin chain. The problem has been attracting a great deal of interest because of its relevance to quantum information processing. Using the Riemann-Hilbert approach, Its, together with V. Korepin and B.-Q. Jin (SUNY at Stony Brook), has evaluated the principal characteristic of the entanglement-the limiting entropy of a block of spins.

In the area of integrable equations, Its is pursuing several long-term projects addressing the global analytic properties of the solutions of integrable equations. One of the most recent of Its' efforts in this area is represented by a series of papers on the Riemann-Hilbert approach to the initial-boundary value problems for nonlinear Schroedinger equation.

Slawomir Klimek

Prof. Klimek research focuses on the noncommutative geometry and its relation to modern theoretical physics. In collaboration with A. Lesniewski, David Borthwick, and Maurizio Rinaldi (Harvard University), he constructed several classes of fundamental examples including quantum Riemann surfaces, Cartan domains, and Dirac operators. He has also obtained a number of results in abstract cyclic cohomology theory in noncommutative spaces.

During recent years, Klimek has been publishing papers on mathematical aspects of quantum field theories, supersymmetry, mathematical quantization, quantum chaos, cyclic theory, eta invariants, infinite dimensional group actions Lie theory, and theory of operators on Banach and Hilbert spaces. He is currently working on concrete projects in quantum chaos, infinite dimensional geometry, and (almost periodic) field theory. Klimek is also actively pursuing the possibilities of applying noncommutative methods in number theory.

Evgeny Mukhin

Prof. Mukhin studies symmetries and structures arising in the context of conformal field theory, quantum field theory and exactly solvable models of statistical physics. He is using a combination of tools from representation theory, combinatorics, and analysis. One of his major contributions is the proofs (together with E. Frenkel of Berkeley) of a number of fundamental conjectures concerning the characters of representations of quantum affine algebras. Although Mukhin is the youngest member of the Mathematical Physics Group, he has an extremely impressive list of external collaborators; one of his long-term collaborators, M. Jimbo (Tokyo, Japan), is a co-founder of the research field of quantum groups.

Mukhin's recent results are concerned, in addition to the theory of representations of the affine quantum groups, with the correlation functions in various models of Conformal Field Theory and with the algebraic Bethe Ansatz. In the near future, he plans to continue his study of correlation functions in minimal models of conformal field theory with relation to the application to the theory of Macdonald polynomials at negative values of Jack coupling constant. A very interesting aspect of these studies is that they link the algebraic Bethe Ansatz to the theory of orthogonal polynomials.

Vitaly Tarasov

Prof. Tarasov works in representation theory of Lie algebras and quantum groups and in the theory of multidimensional hypergeometric functions. Tarasov has made several major contributions in the development of this field. Indeed, his works were used by V. Drinfeld (now of Chicago) when the latter introduced the notion of the Yangians-one of the principal concepts of the theory of quantum groups.

Tarasov's current research interests are concentrated around the differential and difference (quantized) Knizhnik-Zamolodchikov equations, which play a central role in modern theory of exactly solvable quantum field and statistical mechanics models. His research projects concern with the generalizations of the classical Selberg integrals to the higher rank Lie algebras and with the representation theory of the Yangians (a principal object of the quantum group theory mentioned above). Among Tarasov's recent results in these directions are certain highly nontrivial identities between hypergeometric and q-hypergeometric integrals of different dimensions. These identities have attracted strong interest in the special function community because they can be used in some cases to compute asymptotics of hypergeometric integrals with respect to their dimension. Simultaneously, this provides a new direction to attack one of the most challenging problems in the theory of exactly solvable models, which is the asymptotics of the nonfree fermion correlation functions.